Annie Bowman died when she was crushed beneath the wheels of the digger while trying to stop it being driven away.
Police said they may treat her death as murder if the evidence supported it.
The dead woman's godson, Michael Smythe, 31, for whose family Miss Bowman had worked as a housekeeper for 56 years, offered the reward for information about the thief.
He said: 'She came to work for my nan and grandad when she was 16. She was the best woman in my life. Obviously they knew what they had done. They must have such a guilty conscience.'
Miss Bowman was alone in the family's home in Bullsmoor Lane, Enfield, north London, on Thursday night, when she heard someone starting up the digger outside.
She chased the man for 100 yards and fell under the digger's rear wheels, said Det Chief Insp Peter Weston, leading the investigation. Witnesses saw her clinging on as the digger was driven off.
Passing motorists stopped to help her, while others attempted to follow the stolen vehicle until the thief abandoned it in a park and fled. Miss Bowman died later in Chase Farm Hospital, Enfield.
Dave Quelch, owner of a nearby pet-food shop, said Miss Bowman was 'a lovely, gentle lady'. He said a colleague saw her holding on to the side of the JCB before she fell. It did not slow down.
Jimmy Croft, 44, owner of the nearby Waterworld Centre aquarium shop, saw the incident.
He said: 'The JCB went past and there was a figure in green hanging on to the door handle. The figure was hanging on between the two wheels on the left hand side of the vehicle and the driver must have seen her.
'The next thing was that one of my staff looked into the road and saw the figure lying there.'
A neighbour, Samantha Burr, 16, tried to comfort Miss Bowman after the incident. 'She was conscious and was crying. She was saying 'help me, it's hurting'.
'She was a lovely woman. She used to walk the family alsatians about a mile or two every day. She was very strong, not someone who would have crept up behind a JCB. She would have shouted and driver must have seen her. I hope they catch the thief - a JCB is not worth someone's life.'
Scotland Yard said last night the man who drove away the JCB was reported to be white, 'scruffy', about 30, about 5ft 9ins tall, with short or collar-length black hair and probably wearing a black donkey-style jacket.